The Laplace transform is an integral transform that converts a function of time $f(t)$ into a function of a complex variable $s$, denoted as $F(s)$. It is widely used to solve differential equations by transforming them into algebraic equations.
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The Laplace transform of a function $f(t)$ is defined as $F(s) = \int_0^\infty f(t)e^{-st}dt$.
It can simplify the process of solving linear differential equations with given initial conditions.
Common properties include linearity, differentiation in the time domain, and integration in the time domain.
For improper integrals, the convergence of the Laplace transform depends on the behavior of $f(t)$ as $t \to \infty$.
$\mathcal{L}\{1\} = \frac{1}{s}$ for $Re(s) > 0$.
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Related terms
Integral Transform: A mathematical operation that converts a function into another function via integration.